Finite Group Databasebeta

← #41012427

Subgroups — #41012427

Order 27008.

208 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27213743C211yesmaximal, characteristic
2#27213481C211yesmaximal, characteristic, complemented
3#27213464C211yesmaximal, characteristic
4#27212882C211yesmaximal, characteristic, complemented
5#27212879C211yesmaximal, characteristic
6#27212872C211yesmaximal, characteristic, complemented
7#400×C211C211yesmaximal, characteristic, complemented
8#20269972—21no—
9#20269967—21no—
10#20269962—21no—
11#20269960—21no—
12#20269956C2211yescharacteristic
13#20269955—21no—
14#20269951C2211yescharacteristic
15C4×C22⋊D211—21no—
16#20269910—21no—
17#20269908—21no—
18#20269900—21no—
19#20269897—21no—
20C22⋊(C211⋊C8)C2211yescharacteristic
21C22⋊(C211⋊C8)C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24#95×C211—21no—
25#95×C211—21no—
26#95×C211C2211yescharacteristic
27C2×Dic422D411yescharacteristic
28C2×D844D411yescharacteristic
29C2×C22⋊D211—21no—
30C2×C4×D211—21no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C4⋊Dic211D411yescharacteristic
34C4⋊Dic211D411yescharacteristic
35Dic211⋊C4—21no—
36Dic211⋊C4—21no—
37Dic211⋊C4—21no—
38Dic211⋊C4—21no—
39C22⋊Dic211—21no—
40(C2×D211)⋊C4—21no—
41(C2×D211)⋊C4—21no—
42(C2×D211)⋊C4—21no—
43(C2×D211)⋊C4—21no—
44C4×Dic211—21no—
45C22×C4×C211—21no—
46C22×C4×C211—21no—
47C22×C4×C211D411yescharacteristic
48C22×C4×C211D411yescharacteristic
49C22×C4×C211C2311yescharacteristic
50C4⋊C4×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211—21no—
54C22×D211—21no—
55Dic422—41no—
56D844—41no—
57C22⋊D211—41no—
58C22⋊D211—41no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—21no—
62C4×D211—81no—
63C211⋊C8—41no—
64C211⋊C8—41no—
65C23×C211C2×D411yescharacteristic
66C2×C4×C211—21no—
67C2×C4×C211—21no—
68C2×C4×C211—21no—
69C2×C4×C211—21no—
70C2×C4×C211—41no—
71C2×C4×C211—41no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211—42no—
75C2×C4×C211—41no—
76C2×C4×C211—41no—
77C2×C4×C211C2×D411yescharacteristic
78C2×C4×C211C2×D411yescharacteristic
79C22×C211—21no—
80C22×C211—21no—
81C22×C211#12011yescharacteristic
82C22×C211#13511yescharacteristic
83C22×C211#13711yescharacteristic
84C2×D211—41no—
85C2×D211—41no—
86C2×D211—41no—
87C4×C211—21no—
88C4×C211—21no—
89C4×C211—41no—
90C4×C211—41no—
91C4×C211—81no—
92C4×C211—81no—
93Dic211—41no—
94Dic211—41no—
95Dic211—41no—
96C2×C211—21no—
97C2×C211—21no—
98C2×C211#45211yescharacteristic
99C2×C211#45411yescharacteristic
100C2×C211#45811yescharacteristic
101D211—81no—
102C211#161511yescharacteristic, complemented, Sylow
103#1615—2111nomaximal, Sylow
104#539—2111no—
105#484—2111no—
106#481—2111no—
107#400D21111yescharacteristic, complemented
108#328—2111no—
109#327—2111no—
110#324—2111no—
111#126—4221no—
112#125—4221no—
113#123—4221no—
114#123—4221no—
115C22⋊Q8—4221no—
116C22⋊Q8—2111no—
117#121—2111no—
118C4×D4—4221no—
119C2×C4⋊C4C2×D21111yescharacteristic
120C2×C4⋊C4C2×D21111yescharacteristic
121#107—4221no—
122#106—4221no—
123#103—4221no—
124#102—4221no—
125C22⋊C8—2111no—
126C22⋊C8—2111no—
127#95—21no—
128#95—21no—
129#95C2×D21111yescharacteristic
130C2×Q8—2111no—
131C2×D4—4221no—
132C2×D4—2111no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—4221no—
136C22×C4C22⋊D21111yescharacteristic
137C22×C4C22⋊D21111yescharacteristic
138C22×C4C22×D21111yescharacteristic
139C2×C8—4221no—
140C2×C8—4221no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—4221no—
146C4⋊C4—4221no—
147C4⋊C4—4221no—
148C4⋊C4—4221no—
149C4⋊C4—2111no—
150C4⋊C4—2111no—
151C22⋊C4—4221no—
152C22⋊C4—4221no—
153C22⋊C4—4221no—
154C22⋊C4—4221no—
155C22⋊C4—4221no—
156C42—4221no—
157C23—4221no—
158C23C2×C22⋊D21111yescharacteristic
159Q8—8441no—
160D4—8441no—
161D4—8441no—
162D4—8441no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—16881no—
175C2×C4—4221no—
176C2×C4—4221no—
177C2×C4—4221no—
178C2×C4D4×D21111yescharacteristic
179C2×C4D4×D21111yescharacteristic
180C8—8441no—
181C8—8441no—
182C22—21no—
183C22—21no—
184C22—8441no—
185C22—8441no—
186C22—8441no—
187C22#2026994511yescharacteristic
188C22#2026999711yescharacteristic
189C22#2027000811yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—41no—
194C4—81no—
195C4—81no—
196C4—8441no—
197C4—8441no—
198C4—8441no—
199C2—21no—
200C2—21no—
201C2—16881no—
202C2#2721330611yescharacteristic
203C2#2721331811yescharacteristic
204C2#2721333511yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#400×C211C2
socleC22×C211#120
prime radical#20269951C22
hypercenter#400D211
hypocenterC211#1615
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #400

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #27213481C2semidirect
4. #27212882C2semidirect
6. #27212872C2semidirect
7. #400×C211C2semidirect
102. C211#1615semidirect
107. #400D211semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#272137432
#272134812
#272134642
#272128822
#272128792
#272128722
#400×C2112
#1615211